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Proof step 10 from proof at-payload (previous step)

zbrar[m + n1 + 1] = mkzentry(key, adr), m + nused0 < # zbrar, m < # zbrar, n1 < nused0, zbrar[(m + 1) + n1] = mkzentry(key, adr) ∧ n1 < nused0 ∧ (m + 1) + nused0 ≤ # zbrar ∧ (∀ n0. n0 < n1 ∧ zbrar[(m + 1) + n0].entry? → zbrar[(m + 1) + n0].key ≠ key) → payload(zbrar, m + 1, nused0)[key] = adr, ∀ n. ¬ n1 < n ∧ zbrar[m + n].entry? → zbrar[m + n].key ≠ key ⊦ (zbrar[m].entry? ⊃ payload(zbrar, m + 1, nused0)[zbrar[m].key, zbrar[m].adr];payload(zbrar, m + 1, nused0))[key] = adr

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 11:
zbrar[m + n1 + 1] = mkzentry(key, adr), n1 < nused0, m < # zbrar, m + nused0 < # zbrar, (n < n1 ∧ zbrar[(m + 1) + n].entry? → zbrar[(m + 1) + n].key ≠ key) → payload(zbrar, m + 1, nused0)[key] = adr, ∀ n. ¬ n1 < n ∧ zbrar[m + n].entry? → zbrar[m + n].key ≠ key ⊦ (zbrar[m].entry? ⊃ payload(zbrar, m + 1, nused0)[zbrar[m].key, zbrar[m].adr];payload(zbrar, m + 1, nused0))[key] = adr

Rule argument

Additional information

Used simplifier rules: ⊦ m + 1 ≤ n ↔ m < n; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic